Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{4a^3 + 48a^2 + 80a}{a^3 - 5a^2 - 14a}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {4a(a^2 + 12a + 20)} {a(a^2 - 5a - 14)} $ $ z = \dfrac{4a}{a} \cdot \dfrac{a^2 + 12a + 20}{a^2 - 5a - 14} $ Simplify: $ z = 4 \cdot \dfrac{a^2 + 12a + 20}{a^2 - 5a - 14}$ Since we are dividing by $a$ , we must remember that $a \neq 0$ Next factor the numerator and denominator. $ z = 4 \cdot \dfrac{(a + 2)(a + 10)}{(a + 2)(a - 7)}$ Assuming $a \neq -2$ , we can cancel the $a + 2$ $ z = 4 \cdot \dfrac{a + 10}{a - 7}$ Therefore: $ z = \dfrac{ 4(a + 10)}{ a - 7 }$, $a \neq -2$, $a \neq 0$